Find the function $f \cdot g$ and its domain if $f ( x ) = \sqrt { x + 7 }$ and $g ( x ) = \sqrt { x - 7 }$.
A)
$\sqrt { x ^ { 2 } - 49 }$
$D = [ 7 , \infty )$

B) $\sqrt { x ^ { 2 } - 49 }$
$$D = [ - 7 , \infty )$$

C)
$$\begin{array} { l }
\sqrt { x ^ { 2 } + 49 } \
D = [ - 7 , \infty )
\end{array}$$
D) $\sqrt { x ^ { 2 } + 49 }$
$$D = [ 7 , \infty )$$

Question

Find the function $f \cdot g$ and its domain if $f ( x ) = \sqrt { x + 7 }$ and $g ( x ) = \sqrt { x - 7 }$.
A)
 $\sqrt { x ^ { 2 } - 49 }$
$D = [ 7 , \infty )$

B) $\sqrt { x ^ { 2 } - 49 }$
$$D = [ - 7 , \infty )$$

C)
$$\begin{array} { l } 
\sqrt { x ^ { 2 } + 49 } \
D = [ - 7 , \infty )
\end{array}$$
D) $\sqrt { x ^ { 2 } + 49 }$
$$D = [ 7 , \infty )$$

Quizplus · Accepted Answer

To find the product of two functions, we multiply their expressions: $$(f \cdot g)(x) = f(x) \cdot g(x) = \sqrt{x + 7} \cdot \sqrt{x - 7} = \sqrt{(x + 7)(x - 7)} = \sqrt{x^2 - 49}$$ 
However, note that the domain must satisfy the restrictions of both functions being multiplied. The square root of a negative number is undefined in real numbers, so $x \geq 7$ for $f(x)$ and $x \geq 7$ for $g(x)$. Therefore, the domain of the product is $[7, \infty)$. Thus, the answer is A.

Find the Function f⋅gf \cdot gf⋅g And Its Domain If f(x)=x+7f ( x ) = \sqrt { x + 7 }f(x)=x+7​

Find the Function $f \cdot g$ And Its Domain If $f ( x ) = \sqrt { x + 7 }$